Combined (q,h)-Deformation as a Nonlinear Map on $U_q(sl(2))$

Details Combined (q,h)-Deformation as a Nonlinear Map on $U_q(sl(2))$ Combined (q,h)-Deformation as a Nonlinear Map on $U_q(sl(2))$

1996-12-19

arxiv.org/abs/q-alg/9612028v1

Mathematics

Quantum Algebra

Latex, 11 pages

Scientific paper

The generators $(J_{\pm}, J_0)$ of the algebra $U_q(sl(2))$ is our starting point. An invertible nonlinear map involving, apart from q, a second arbitrary complex parameter h, defines a triplet $({\hat X},{\hat Y},{\hat H})$. The latter set forms a closed algebra under commutation relations. The nonlinear algebra $U_{q,h}(sl(2))$, thus generated, has two different limits. For $q \to 1$, the Jordanian h-deformation $U_{h}(sl(2))$ is obtained. For $h \to 0$, the q-deformed algebra $U_{q}(sl(2))$ is reproduced. From the nonlinear map, the irreducible representations of the doubly-deformed algebra $U_{q,h}(sl(2))$ may be directly and explicitly obtained form the known representations of the algebra $U_q(sl(2))$. Here we consider only generic values of q.

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